3 * Implementation of class for extended truncated power series and
4 * methods for series expansion. */
7 * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
9 * This program is free software; you can redistribute it and/or modify
10 * it under the terms of the GNU General Public License as published by
11 * the Free Software Foundation; either version 2 of the License, or
12 * (at your option) any later version.
14 * This program is distributed in the hope that it will be useful,
15 * but WITHOUT ANY WARRANTY; without even the implied warranty of
16 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
17 * GNU General Public License for more details.
19 * You should have received a copy of the GNU General Public License
20 * along with this program; if not, write to the Free Software
21 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
29 #include "inifcns.h" // for Order function
33 #include "relational.h"
34 #include "operators.h"
41 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(pseries, basic,
42 print_func<print_context>(&pseries::do_print).
43 print_func<print_latex>(&pseries::do_print_latex).
44 print_func<print_tree>(&pseries::do_print_tree).
45 print_func<print_python>(&pseries::do_print_python).
46 print_func<print_python_repr>(&pseries::do_print_python_repr))
53 pseries::pseries() : inherited(TINFO_pseries) { }
60 /** Construct pseries from a vector of coefficients and powers.
61 * expair.rest holds the coefficient, expair.coeff holds the power.
62 * The powers must be integers (positive or negative) and in ascending order;
63 * the last coefficient can be Order(_ex1) to represent a truncated,
64 * non-terminating series.
66 * @param rel_ expansion variable and point (must hold a relational)
67 * @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero)
68 * @return newly constructed pseries */
69 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
71 GINAC_ASSERT(is_a<relational>(rel_));
72 GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
82 pseries::pseries(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
84 for (unsigned int i=0; true; ++i) {
87 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
88 seq.push_back(expair(rest, coeff));
92 n.find_ex("var", var, sym_lst);
93 n.find_ex("point", point, sym_lst);
96 void pseries::archive(archive_node &n) const
98 inherited::archive(n);
99 epvector::const_iterator i = seq.begin(), iend = seq.end();
101 n.add_ex("coeff", i->rest);
102 n.add_ex("power", i->coeff);
105 n.add_ex("var", var);
106 n.add_ex("point", point);
109 DEFAULT_UNARCHIVE(pseries)
112 // functions overriding virtual functions from base classes
115 void pseries::print_series(const print_context & c, const char *openbrace, const char *closebrace, const char *mul_sym, const char *pow_sym, unsigned level) const
117 if (precedence() <= level)
120 // objects of type pseries must not have any zero entries, so the
121 // trivial (zero) pseries needs a special treatment here:
125 epvector::const_iterator i = seq.begin(), end = seq.end();
128 // print a sign, if needed
129 if (i != seq.begin())
132 if (!is_order_function(i->rest)) {
134 // print 'rest', i.e. the expansion coefficient
135 if (i->rest.info(info_flags::numeric) &&
136 i->rest.info(info_flags::positive)) {
139 c.s << openbrace << '(';
141 c.s << ')' << closebrace;
144 // print 'coeff', something like (x-1)^42
145 if (!i->coeff.is_zero()) {
147 if (!point.is_zero()) {
148 c.s << openbrace << '(';
149 (var-point).print(c);
150 c.s << ')' << closebrace;
153 if (i->coeff.compare(_ex1)) {
156 if (i->coeff.info(info_flags::negative)) {
166 Order(power(var-point,i->coeff)).print(c);
170 if (precedence() <= level)
174 void pseries::do_print(const print_context & c, unsigned level) const
176 print_series(c, "", "", "*", "^", level);
179 void pseries::do_print_latex(const print_latex & c, unsigned level) const
181 print_series(c, "{", "}", " ", "^", level);
184 void pseries::do_print_python(const print_python & c, unsigned level) const
186 print_series(c, "", "", "*", "**", level);
189 void pseries::do_print_tree(const print_tree & c, unsigned level) const
191 c.s << std::string(level, ' ') << class_name() << " @" << this
192 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
194 size_t num = seq.size();
195 for (size_t i=0; i<num; ++i) {
196 seq[i].rest.print(c, level + c.delta_indent);
197 seq[i].coeff.print(c, level + c.delta_indent);
198 c.s << std::string(level + c.delta_indent, ' ') << "-----" << std::endl;
200 var.print(c, level + c.delta_indent);
201 point.print(c, level + c.delta_indent);
204 void pseries::do_print_python_repr(const print_python_repr & c, unsigned level) const
206 c.s << class_name() << "(relational(";
211 size_t num = seq.size();
212 for (size_t i=0; i<num; ++i) {
216 seq[i].rest.print(c);
218 seq[i].coeff.print(c);
224 int pseries::compare_same_type(const basic & other) const
226 GINAC_ASSERT(is_a<pseries>(other));
227 const pseries &o = static_cast<const pseries &>(other);
229 // first compare the lengths of the series...
230 if (seq.size()>o.seq.size())
232 if (seq.size()<o.seq.size())
235 // ...then the expansion point...
236 int cmpval = var.compare(o.var);
239 cmpval = point.compare(o.point);
243 // ...and if that failed the individual elements
244 epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
245 while (it!=seq.end() && o_it!=o.seq.end()) {
246 cmpval = it->compare(*o_it);
253 // so they are equal.
257 /** Return the number of operands including a possible order term. */
258 size_t pseries::nops() const
263 /** Return the ith term in the series when represented as a sum. */
264 ex pseries::op(size_t i) const
267 throw (std::out_of_range("op() out of range"));
269 return seq[i].rest * power(var - point, seq[i].coeff);
272 /** Return degree of highest power of the series. This is usually the exponent
273 * of the Order term. If s is not the expansion variable of the series, the
274 * series is examined termwise. */
275 int pseries::degree(const ex &s) const
277 if (var.is_equal(s)) {
278 // Return last exponent
280 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
284 epvector::const_iterator it = seq.begin(), itend = seq.end();
287 int max_pow = INT_MIN;
288 while (it != itend) {
289 int pow = it->rest.degree(s);
298 /** Return degree of lowest power of the series. This is usually the exponent
299 * of the leading term. If s is not the expansion variable of the series, the
300 * series is examined termwise. If s is the expansion variable but the
301 * expansion point is not zero the series is not expanded to find the degree.
302 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
303 int pseries::ldegree(const ex &s) const
305 if (var.is_equal(s)) {
306 // Return first exponent
308 return ex_to<numeric>((seq.begin())->coeff).to_int();
312 epvector::const_iterator it = seq.begin(), itend = seq.end();
315 int min_pow = INT_MAX;
316 while (it != itend) {
317 int pow = it->rest.ldegree(s);
326 /** Return coefficient of degree n in power series if s is the expansion
327 * variable. If the expansion point is nonzero, by definition the n=1
328 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
329 * the expansion took place in the s in the first place).
330 * If s is not the expansion variable, an attempt is made to convert the
331 * series to a polynomial and return the corresponding coefficient from
333 ex pseries::coeff(const ex &s, int n) const
335 if (var.is_equal(s)) {
339 // Binary search in sequence for given power
340 numeric looking_for = numeric(n);
341 int lo = 0, hi = seq.size() - 1;
343 int mid = (lo + hi) / 2;
344 GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
345 int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
351 return seq[mid].rest;
356 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
361 return convert_to_poly().coeff(s, n);
365 ex pseries::collect(const ex &s, bool distributed) const
370 /** Perform coefficient-wise automatic term rewriting rules in this class. */
371 ex pseries::eval(int level) const
376 if (level == -max_recursion_level)
377 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
379 // Construct a new series with evaluated coefficients
381 new_seq.reserve(seq.size());
382 epvector::const_iterator it = seq.begin(), itend = seq.end();
383 while (it != itend) {
384 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
387 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
390 /** Evaluate coefficients numerically. */
391 ex pseries::evalf(int level) const
396 if (level == -max_recursion_level)
397 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
399 // Construct a new series with evaluated coefficients
401 new_seq.reserve(seq.size());
402 epvector::const_iterator it = seq.begin(), itend = seq.end();
403 while (it != itend) {
404 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
407 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
410 ex pseries::subs(const exmap & m, unsigned options) const
412 // If expansion variable is being substituted, convert the series to a
413 // polynomial and do the substitution there because the result might
414 // no longer be a power series
415 if (m.find(var) != m.end())
416 return convert_to_poly(true).subs(m, options);
418 // Otherwise construct a new series with substituted coefficients and
421 newseq.reserve(seq.size());
422 epvector::const_iterator it = seq.begin(), itend = seq.end();
423 while (it != itend) {
424 newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
427 return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
430 /** Implementation of ex::expand() for a power series. It expands all the
431 * terms individually and returns the resulting series as a new pseries. */
432 ex pseries::expand(unsigned options) const
435 epvector::const_iterator i = seq.begin(), end = seq.end();
437 ex restexp = i->rest.expand();
438 if (!restexp.is_zero())
439 newseq.push_back(expair(restexp, i->coeff));
442 return (new pseries(relational(var,point), newseq))
443 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
446 /** Implementation of ex::diff() for a power series.
448 ex pseries::derivative(const symbol & s) const
451 epvector::const_iterator it = seq.begin(), itend = seq.end();
455 // FIXME: coeff might depend on var
456 while (it != itend) {
457 if (is_order_function(it->rest)) {
458 new_seq.push_back(expair(it->rest, it->coeff - 1));
460 ex c = it->rest * it->coeff;
462 new_seq.push_back(expair(c, it->coeff - 1));
469 while (it != itend) {
470 if (is_order_function(it->rest)) {
471 new_seq.push_back(*it);
473 ex c = it->rest.diff(s);
475 new_seq.push_back(expair(c, it->coeff));
481 return pseries(relational(var,point), new_seq);
484 ex pseries::convert_to_poly(bool no_order) const
487 epvector::const_iterator it = seq.begin(), itend = seq.end();
489 while (it != itend) {
490 if (is_order_function(it->rest)) {
492 e += Order(power(var - point, it->coeff));
494 e += it->rest * power(var - point, it->coeff);
500 bool pseries::is_terminating() const
502 return seq.empty() || !is_order_function((seq.end()-1)->rest);
507 * Implementations of series expansion
510 /** Default implementation of ex::series(). This performs Taylor expansion.
512 ex basic::series(const relational & r, int order, unsigned options) const
517 ex coeff = deriv.subs(r, subs_options::no_pattern);
518 const symbol &s = ex_to<symbol>(r.lhs());
520 if (!coeff.is_zero())
521 seq.push_back(expair(coeff, _ex0));
524 for (n=1; n<order; ++n) {
526 // We need to test for zero in order to see if the series terminates.
527 // The problem is that there is no such thing as a perfect test for
528 // zero. Expanding the term occasionally helps a little...
529 deriv = deriv.diff(s).expand();
530 if (deriv.is_zero()) // Series terminates
531 return pseries(r, seq);
533 coeff = deriv.subs(r, subs_options::no_pattern);
534 if (!coeff.is_zero())
535 seq.push_back(expair(fac.inverse() * coeff, n));
538 // Higher-order terms, if present
539 deriv = deriv.diff(s);
540 if (!deriv.expand().is_zero())
541 seq.push_back(expair(Order(_ex1), n));
542 return pseries(r, seq);
546 /** Implementation of ex::series() for symbols.
548 ex symbol::series(const relational & r, int order, unsigned options) const
551 const ex point = r.rhs();
552 GINAC_ASSERT(is_a<symbol>(r.lhs()));
554 if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
555 if (order > 0 && !point.is_zero())
556 seq.push_back(expair(point, _ex0));
558 seq.push_back(expair(_ex1, _ex1));
560 seq.push_back(expair(Order(_ex1), numeric(order)));
562 seq.push_back(expair(*this, _ex0));
563 return pseries(r, seq);
567 /** Add one series object to another, producing a pseries object that
568 * represents the sum.
570 * @param other pseries object to add with
571 * @return the sum as a pseries */
572 ex pseries::add_series(const pseries &other) const
574 // Adding two series with different variables or expansion points
575 // results in an empty (constant) series
576 if (!is_compatible_to(other)) {
578 nul.push_back(expair(Order(_ex1), _ex0));
579 return pseries(relational(var,point), nul);
584 epvector::const_iterator a = seq.begin();
585 epvector::const_iterator b = other.seq.begin();
586 epvector::const_iterator a_end = seq.end();
587 epvector::const_iterator b_end = other.seq.end();
588 int pow_a = INT_MAX, pow_b = INT_MAX;
590 // If a is empty, fill up with elements from b and stop
593 new_seq.push_back(*b);
598 pow_a = ex_to<numeric>((*a).coeff).to_int();
600 // If b is empty, fill up with elements from a and stop
603 new_seq.push_back(*a);
608 pow_b = ex_to<numeric>((*b).coeff).to_int();
610 // a and b are non-empty, compare powers
612 // a has lesser power, get coefficient from a
613 new_seq.push_back(*a);
614 if (is_order_function((*a).rest))
617 } else if (pow_b < pow_a) {
618 // b has lesser power, get coefficient from b
619 new_seq.push_back(*b);
620 if (is_order_function((*b).rest))
624 // Add coefficient of a and b
625 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
626 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
627 break; // Order term ends the sequence
629 ex sum = (*a).rest + (*b).rest;
630 if (!(sum.is_zero()))
631 new_seq.push_back(expair(sum, numeric(pow_a)));
637 return pseries(relational(var,point), new_seq);
641 /** Implementation of ex::series() for sums. This performs series addition when
642 * adding pseries objects.
644 ex add::series(const relational & r, int order, unsigned options) const
646 ex acc; // Series accumulator
648 // Get first term from overall_coeff
649 acc = overall_coeff.series(r, order, options);
651 // Add remaining terms
652 epvector::const_iterator it = seq.begin();
653 epvector::const_iterator itend = seq.end();
654 for (; it!=itend; ++it) {
656 if (is_exactly_a<pseries>(it->rest))
659 op = it->rest.series(r, order, options);
660 if (!it->coeff.is_equal(_ex1))
661 op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
664 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
670 /** Multiply a pseries object with a numeric constant, producing a pseries
671 * object that represents the product.
673 * @param other constant to multiply with
674 * @return the product as a pseries */
675 ex pseries::mul_const(const numeric &other) const
678 new_seq.reserve(seq.size());
680 epvector::const_iterator it = seq.begin(), itend = seq.end();
681 while (it != itend) {
682 if (!is_order_function(it->rest))
683 new_seq.push_back(expair(it->rest * other, it->coeff));
685 new_seq.push_back(*it);
688 return pseries(relational(var,point), new_seq);
692 /** Multiply one pseries object to another, producing a pseries object that
693 * represents the product.
695 * @param other pseries object to multiply with
696 * @return the product as a pseries */
697 ex pseries::mul_series(const pseries &other) const
699 // Multiplying two series with different variables or expansion points
700 // results in an empty (constant) series
701 if (!is_compatible_to(other)) {
703 nul.push_back(expair(Order(_ex1), _ex0));
704 return pseries(relational(var,point), nul);
707 // Series multiplication
709 int a_max = degree(var);
710 int b_max = other.degree(var);
711 int a_min = ldegree(var);
712 int b_min = other.ldegree(var);
713 int cdeg_min = a_min + b_min;
714 int cdeg_max = a_max + b_max;
716 int higher_order_a = INT_MAX;
717 int higher_order_b = INT_MAX;
718 if (is_order_function(coeff(var, a_max)))
719 higher_order_a = a_max + b_min;
720 if (is_order_function(other.coeff(var, b_max)))
721 higher_order_b = b_max + a_min;
722 int higher_order_c = std::min(higher_order_a, higher_order_b);
723 if (cdeg_max >= higher_order_c)
724 cdeg_max = higher_order_c - 1;
726 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
728 // c(i)=a(0)b(i)+...+a(i)b(0)
729 for (int i=a_min; cdeg-i>=b_min; ++i) {
730 ex a_coeff = coeff(var, i);
731 ex b_coeff = other.coeff(var, cdeg-i);
732 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
733 co += a_coeff * b_coeff;
736 new_seq.push_back(expair(co, numeric(cdeg)));
738 if (higher_order_c < INT_MAX)
739 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
740 return pseries(relational(var, point), new_seq);
744 /** Implementation of ex::series() for product. This performs series
745 * multiplication when multiplying series.
747 ex mul::series(const relational & r, int order, unsigned options) const
749 pseries acc; // Series accumulator
751 // Multiply with remaining terms
752 const epvector::const_iterator itbeg = seq.begin();
753 const epvector::const_iterator itend = seq.end();
754 for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
755 ex op = recombine_pair_to_ex(*it).series(r, order, options);
757 // Series multiplication
759 acc = ex_to<pseries>(op);
761 acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
763 return acc.mul_const(ex_to<numeric>(overall_coeff));
767 /** Compute the p-th power of a series.
769 * @param p power to compute
770 * @param deg truncation order of series calculation */
771 ex pseries::power_const(const numeric &p, int deg) const
774 // (due to Leonhard Euler)
775 // let A(x) be this series and for the time being let it start with a
776 // constant (later we'll generalize):
777 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
778 // We want to compute
780 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
781 // Taking the derivative on both sides and multiplying with A(x) one
782 // immediately arrives at
783 // C'(x)*A(x) = p*C(x)*A'(x)
784 // Multiplying this out and comparing coefficients we get the recurrence
786 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
787 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
788 // which can easily be solved given the starting value c_0 = (a_0)^p.
789 // For the more general case where the leading coefficient of A(x) is not
790 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
791 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
792 // then of course x^(p*m) but the recurrence formula still holds.
795 // as a special case, handle the empty (zero) series honoring the
796 // usual power laws such as implemented in power::eval()
797 if (p.real().is_zero())
798 throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
799 else if (p.real().is_negative())
800 throw pole_error("pseries::power_const(): division by zero",1);
805 const int ldeg = ldegree(var);
806 if (!(p*ldeg).is_integer())
807 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
809 // O(x^n)^(-m) is undefined
810 if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
811 throw pole_error("pseries::power_const(): division by zero",1);
813 // Compute coefficients of the powered series
816 co.push_back(power(coeff(var, ldeg), p));
817 bool all_sums_zero = true;
818 for (int i=1; i<deg; ++i) {
820 for (int j=1; j<=i; ++j) {
821 ex c = coeff(var, j + ldeg);
822 if (is_order_function(c)) {
823 co.push_back(Order(_ex1));
826 sum += (p * j - (i - j)) * co[i - j] * c;
829 all_sums_zero = false;
830 co.push_back(sum / coeff(var, ldeg) / i);
833 // Construct new series (of non-zero coefficients)
835 bool higher_order = false;
836 for (int i=0; i<deg; ++i) {
837 if (!co[i].is_zero())
838 new_seq.push_back(expair(co[i], p * ldeg + i));
839 if (is_order_function(co[i])) {
844 if (!higher_order && !all_sums_zero)
845 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
846 return pseries(relational(var,point), new_seq);
850 /** Return a new pseries object with the powers shifted by deg. */
851 pseries pseries::shift_exponents(int deg) const
853 epvector newseq = seq;
854 epvector::iterator i = newseq.begin(), end = newseq.end();
859 return pseries(relational(var, point), newseq);
863 /** Implementation of ex::series() for powers. This performs Laurent expansion
864 * of reciprocals of series at singularities.
866 ex power::series(const relational & r, int order, unsigned options) const
868 // If basis is already a series, just power it
869 if (is_exactly_a<pseries>(basis))
870 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
872 // Basis is not a series, may there be a singularity?
873 bool must_expand_basis = false;
875 basis.subs(r, subs_options::no_pattern);
876 } catch (pole_error) {
877 must_expand_basis = true;
880 // Is the expression of type something^(-int)?
881 if (!must_expand_basis && !exponent.info(info_flags::negint))
882 return basic::series(r, order, options);
884 // Is the expression of type 0^something?
885 if (!must_expand_basis && !basis.subs(r, subs_options::no_pattern).is_zero())
886 return basic::series(r, order, options);
888 // Singularity encountered, is the basis equal to (var - point)?
889 if (basis.is_equal(r.lhs() - r.rhs())) {
891 if (ex_to<numeric>(exponent).to_int() < order)
892 new_seq.push_back(expair(_ex1, exponent));
894 new_seq.push_back(expair(Order(_ex1), exponent));
895 return pseries(r, new_seq);
898 // No, expand basis into series
899 ex e = basis.series(r, order, options);
900 return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
904 /** Re-expansion of a pseries object. */
905 ex pseries::series(const relational & r, int order, unsigned options) const
907 const ex p = r.rhs();
908 GINAC_ASSERT(is_a<symbol>(r.lhs()));
909 const symbol &s = ex_to<symbol>(r.lhs());
911 if (var.is_equal(s) && point.is_equal(p)) {
912 if (order > degree(s))
916 epvector::const_iterator it = seq.begin(), itend = seq.end();
917 while (it != itend) {
918 int o = ex_to<numeric>(it->coeff).to_int();
920 new_seq.push_back(expair(Order(_ex1), o));
923 new_seq.push_back(*it);
926 return pseries(r, new_seq);
929 return convert_to_poly().series(r, order, options);
933 /** Compute the truncated series expansion of an expression.
934 * This function returns an expression containing an object of class pseries
935 * to represent the series. If the series does not terminate within the given
936 * truncation order, the last term of the series will be an order term.
938 * @param r expansion relation, lhs holds variable and rhs holds point
939 * @param order truncation order of series calculations
940 * @param options of class series_options
941 * @return an expression holding a pseries object */
942 ex ex::series(const ex & r, int order, unsigned options) const
947 if (is_a<relational>(r))
948 rel_ = ex_to<relational>(r);
949 else if (is_a<symbol>(r))
950 rel_ = relational(r,_ex0);
952 throw (std::logic_error("ex::series(): expansion point has unknown type"));
955 e = bp->series(rel_, order, options);
956 } catch (std::exception &x) {
957 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));